Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-7x+9y &= -2 \\ -x+3y &= -6\end{align*}$
We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $3$ $\begin{align*}7x-9y &= 2\\ -3x+9y &= -18\end{align*}$ Add the top and bottom equations. $4x = -16$ Divide both sides by $4$ and reduce as necessary. $x = -4$ Substitute $-4$ for $x$ in the top equation. $-7( -4)+9y = -2$ $28+9y = -2$ $9y = -30$ $y = -\dfrac{10}{3}$ The solution is $\enspace x = -4, \enspace y = -\dfrac{10}{3}$.